Muon replaces a matrix gradient by an approximate polar factor and has shown strong empirical performance in neural-network training. The Isotropic Curvature Model (ICM) provides a curvature-based explanation for this update by asking when gradient orthogonalization minimizes a one-step local-loss surrogate. We investigate the stability of this explanation when the fixed input radius of the ICM is replaced by a general radial law. We introduce the Random-Radius Isotropic Curvature Model (RR-ICM) and the Polar-Factor Admissibility Set, which encodes exact scaled-polar optimality as a geometric condition on the positive polar factor of the gradient. We derive an exact characterization of this set for arbitrary radial laws and show that all nonscalar admissible responses arise from radial atoms aligned with curvature kinks. Consequently, every atomless radial law exhibits zero-capacity rigidity: exact scaled-polar optimality is possible only when the gradient is already a scaled column isometry. To connect the curvature-based framework with the established spectral geometry of Muon, we introduce the Minimax RR-ICM by replacing directional averaging with worst-case optimization over probability laws on the sphere. We prove that every nonzero global minimizer lies on the polar ray of the gradient, while its scale is determined by a one-dimensional convex problem. The corresponding admissibility geometry is a positive-semidefinite trace slab, in sharp contrast to the spherical-zonoid geometry of the RR-ICM. Together, these results expose the radial fragility of average-curvature explanations and provide a new minimax variational characterization of Muon's polar direction.
Figures & tables
Figure 1: Radial averaging is a small perturbation at the level of function values but a singular perturbation at a kink. For H(t)=(t−1)+ and Rε∼Unif[1−ε,1+ε] , the averaged profile Hε(t)=E(Rεt−1)+ differs from H only within an O(ε) transition layer. Nevertheless, the interval ∂H(1)=[0,1] is replaced by the unique derivative Hε′(1)=1/2+ε/4 .
Figure 2: Two polar-factor admissibility geometries in dimension n=2 . Panel (a) shows the RR-ICM admissibility set bcI2+wcK2 in spectral coordinates. Its nonscalar extent is controlled by the atom–kink response capacity wc and collapses to the singleton {bcI2} when wc=0 . Panel (b) shows the Minimax RR-ICM admissibility set, where α−:=min∂HR(c) and α+:=max∂HR(c) . Positive semidefiniteness confines the spectrum to the first quadrant, while trace matching restricts it to the slab α−≤λ1+λ2≤α+ . Thus, the RR-ICM restricts spectral shape through spherical-zonoid geometry, whereas the Minimax RR-ICM restricts only the total spectral mass.
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